Which of the following two propositions can neither be both true together nor…

Concept

In traditional categorical logic, propositions are classified as A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative).

Contradictory pairs are A–O and E–I. Exactly one member of a contradictory pair is true, so its members can neither both be true nor both be false.

Application

1. ‘All squares are rectangles’ is an A-type proposition, while ‘Some squares are not rectangles’ is an O-type proposition. Together they form the A–O contradictory pair.

2. ‘Some squares are rectangles’ is an I-type proposition, while ‘No squares are rectangles’ is an E-type proposition. Together they form the E–I contradictory pair.

3. Among the offered pairs, ‘All squares are rectangles’ with ‘Some squares are not rectangles’ is the A–O pair.

Cross-check and contrast

• ‘Some squares are rectangles’ with ‘Some squares are not rectangles’ combines I and O, the two particular propositions; they may both be true.

• ‘No squares are rectangles’ with ‘Some squares are not rectangles’ combines E and O; under the traditional square, truth flows from E to O, so they may both be true.

• ‘All squares are rectangles’ with ‘No squares are rectangles’ combines A and E, the contrary propositions; they cannot both be true, but they may both be false.

Result

Therefore, the required pair is ‘All squares are rectangles’ and ‘Some squares are not rectangles’.